Recently [Konyagin and Shkredov improved](https://arxiv.org/abs/1602.03473) the exponent of $4/3$ in the sum-products estimate in $\mathbb{R}$, namely that $|A+A|+|A\cdot A|\gg |A|^{4/3-o(1)}$ for every $A\subset \mathbb{R}$, to $4/3+5/9813$. This appears to be much harder than the short proof for $4/3-o(1)$ by [Solymosi][1]. 

The conjecture of Erdős is that the exponent approaches 2.


  [1]: https://arxiv.org/pdf/0806.1040.pdf